Remark 2.7 
The interior weak and strong Harnack's inequalities for bounded solutions of
Eq. (REF ) under hypotheses \(({\rm g}_{0})\) –\(({\rm g}_{3})\) 
were established in [1]} (see
also [2]}, [3]} in the context of
generalized Orlicz growth and logarithmic condition).
But unlike [2]}, [3]}, the traditional Moser method
[6]}, [7]} is inapplicable for the proof of Theorem REF ,
since checking whether
the logarithm of solutions belongs to the BMO space is a problem for the non-logarithmic condition
(REF ) with unbounded \(\lambda (r)\) .
We use Trudinger's arguments [8]} adapted to Eq. (REF )
near the boundary \(\partial \Omega \)  (see Section ).

